<!DOCTYPE html>
<html class="writer-html5" lang="en" >
<head>
    <meta charset="utf-8" />
    <meta http-equiv="X-UA-Compatible" content="IE=edge" />
    <meta name="viewport" content="width=device-width, initial-scale=1.0" />
      <link rel="shortcut icon" href="../../img/favicon.ico" />
    <title>连续时间傅里叶变换 - 咩咩的笔记</title>
    <link rel="stylesheet" href="../../css/theme.css" />
    <link rel="stylesheet" href="../../css/theme_extra.css" />
        <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/10.5.0/styles/github.min.css" />
    
      <script>
        // Current page data
        var mkdocs_page_name = "\u8fde\u7eed\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362";
        var mkdocs_page_input_path = "\u4fe1\u53f7\u4e0e\u7cfb\u7edf\\4. \u8fde\u7eed\u65f6\u95f4\u5085\u91cc\u53f6\u53d8\u6362.md";
        var mkdocs_page_url = null;
      </script>
    
    <script src="../../js/jquery-3.6.0.min.js" defer></script>
    <!--[if lt IE 9]>
      <script src="../../js/html5shiv.min.js"></script>
    <![endif]-->
      <script src="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/10.5.0/highlight.min.js"></script>
      <script>hljs.initHighlightingOnLoad();</script> 
</head>

<body class="wy-body-for-nav" role="document">

  <div class="wy-grid-for-nav">
    <nav data-toggle="wy-nav-shift" class="wy-nav-side stickynav">
    <div class="wy-side-scroll">
      <div class="wy-side-nav-search">
          <a href="../.." class="icon icon-home"> 咩咩的笔记
        </a><div role="search">
  <form id ="rtd-search-form" class="wy-form" action="../../search.html" method="get">
      <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" title="Type search term here" />
  </form>
</div>
      </div>

      <div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu">
              <ul>
                <li class="toctree-l1"><a class="reference internal" href="../..">主页</a>
                </li>
              </ul>
              <p class="caption"><span class="caption-text">笔记</span></p>
              <ul class="current">
                  <li class="toctree-l1"><a class="reference internal" href="#">线性代数</a>
    <ul>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/0-%E5%89%8D%E8%A8%80/">0-前言</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/1-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/">1-线性方程组</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/2-%E7%9F%A9%E9%98%B5%E4%BB%A3%E6%95%B0/">2-矩阵代数</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/3-%E8%A1%8C%E5%88%97%E5%BC%8F/">3-行列式</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/4-%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4/">4-向量空间</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/5-%E7%89%B9%E5%BE%81%E5%80%BC%E4%B8%8E%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F/">5-特征值与特征向量</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/6-%E6%AD%A3%E4%BA%A4%E6%80%A7%E4%B8%8E%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98/">6-正交性与最小二乘</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/7-%E5%AF%B9%E7%A7%B0%E9%98%B5%E4%B8%8E%E4%BA%8C%E6%AC%A1%E5%9E%8B/">7-对称阵与二次型</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/8-%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4%E7%9A%84%E5%87%A0%E4%BD%95/">8-向量空间的几何</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E9%99%84%E5%BD%95A-3Blue1Brown%E7%AC%94%E8%AE%B0/">附录A-3Blue1Brown笔记</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E9%99%84%E5%BD%95B-%E9%9B%B6%E7%A9%BA%E9%97%B4%E4%B8%8E%E5%88%97%E7%A9%BA%E9%97%B4%E7%9A%84%E5%AF%B9%E6%AF%94/">附录B-零空间与列空间的对比</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E9%99%84%E5%BD%95C-%E9%80%86%E7%9F%A9%E9%98%B5%E5%AE%9A%E7%90%86/">附录C-逆矩阵定理</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E9%99%84%E5%BD%95D-%E6%80%9D%E7%BB%B4%E5%AF%BC%E5%9B%BE/">附录D-思维导图</a>
                </li>
    </ul>
                  </li>
                  <li class="toctree-l1"><a class="reference internal" href="#">数字电路</a>
    <ul>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/1.%20%E4%BB%8B%E7%BB%8D/">介绍</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/2.%20%E6%95%B0%E5%AD%97%E7%B3%BB%E7%BB%9F%E3%80%81%E8%BF%90%E7%AE%97%E5%92%8C%E7%BC%96%E7%A0%81/">数字系统、运算和编码</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/3.%20%E9%80%BB%E8%BE%91%E9%97%A8/">逻辑门</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/4.%20%E5%B8%83%E5%B0%94%E4%BB%A3%E6%95%B0/">布尔代数</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/5.%20%E7%BB%84%E5%90%88%E9%80%BB%E8%BE%91%E5%88%86%E6%9E%90/">组合逻辑分析</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/6.%20%E7%BB%84%E5%90%88%E9%80%BB%E8%BE%91%E5%8A%9F%E8%83%BD%E6%A8%A1%E5%9D%97/">组合逻辑功能模块</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/7.%20%E9%94%81%E5%AD%98%E5%99%A8%E3%80%81%E8%A7%A6%E5%8F%91%E5%99%A8%E5%92%8C%E5%AE%9A%E6%97%B6%E5%99%A8/">锁存器、触发器和定时器</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/8.%20%E7%A7%BB%E4%BD%8D%E5%AF%84%E5%AD%98%E5%99%A8/">移位寄存器</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/9.%20%E8%AE%A1%E6%95%B0%E5%99%A8/">计数器</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/10.%20%E5%82%A8%E5%AD%98%E5%99%A8/">储存器</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%95%B0%E5%AD%97%E7%94%B5%E8%B7%AF/11.%20%E6%A8%A1%E6%95%B0%E8%BD%AC%E6%8D%A2/">模数转换</a>
                </li>
    </ul>
                  </li>
                  <li class="toctree-l1"><a class="reference internal" href="#">离散数学</a>
    <ul>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/2-%E5%91%BD%E9%A2%98%E9%80%BB%E8%BE%91/">命题逻辑</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/3-%E4%B8%80%E9%98%B6%E9%80%BB%E8%BE%91/">一阶逻辑</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/4-%E8%AF%81%E6%98%8E%E6%96%B9%E6%B3%95/">证明方法</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/5-%E9%9B%86%E5%90%88/">集合</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/6-%E5%85%B3%E7%B3%BB/">关系</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/7-%E5%87%BD%E6%95%B0/">函数</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/8-%E8%AE%A1%E6%95%B0%E4%B8%8E%E7%BB%84%E5%90%88/">计数与组合</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/9-%E5%9B%BE%E4%B8%8E%E6%A0%91/">图与树</a>
                </li>
    </ul>
                  </li>
                  <li class="toctree-l1"><a class="reference internal" href="#">计算机组成原理</a>
    <ul>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/1.%20%E8%AE%A1%E7%AE%97%E6%9C%BA%E6%A6%82%E8%A7%88/">计算机概览</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/2.%20%E6%8C%87%E4%BB%A4/">指令</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/3.%20%E8%AE%A1%E7%AE%97%E6%9C%BA%E4%B8%AD%E7%9A%84%E8%BF%90%E7%AE%97/">计算机中的运算</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/4.%20MIPS%20CPU%E8%AE%BE%E8%AE%A1/">MIPS CPU设计</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/5.%20%E5%AD%98%E5%82%A8%E5%99%A8%E5%B1%82%E6%AC%A1%E7%BB%93%E6%9E%84/">存储器层次结构</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/6.%20%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%80%BB%E7%BA%BF/">计算机系统总线</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/7.%20%E8%BE%93%E5%85%A5%E8%BE%93%E5%87%BA%E7%B3%BB%E7%BB%9F/">输入输出系统</a>
                </li>
    </ul>
                  </li>
                  <li class="toctree-l1"><a class="reference internal" href="#">计算机组成原理实验</a>
    <ul>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/1/1/">加法器</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/2/2/">有限状态机</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/3/3/">MIPS指令集1</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/4/4/">MIPS指令集2</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/5/5/">存储器实验</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/6/6/">寄存器堆与 ALU 设计实验</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/7/7/">存储器与控制器实验</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/8/8/">单周期处理器实验</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/9/9/">多周期处理器实验</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/10/10/">多周期处理器综合性开放实验</a>
                </li>
    </ul>
                  </li>
                  <li class="toctree-l1"><a class="reference internal" href="#">概率论</a>
    <ul>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%A6%82%E7%8E%87%E8%AE%BA/1.%20%E6%A6%82%E7%8E%87%E8%AE%BA%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/">概率论的基本概念</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%A6%82%E7%8E%87%E8%AE%BA/2.%20%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%8F%8A%E5%85%B6%E5%88%86%E5%B8%83/">随机变量及其分布</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%A6%82%E7%8E%87%E8%AE%BA/3.%20%E5%A4%9A%E7%BB%B4%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%8F%8A%E5%85%B6%E5%88%86%E5%B8%83/">多维随机变量及其分布</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%A6%82%E7%8E%87%E8%AE%BA/4.%20%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E7%9A%84%E6%95%B0%E5%AD%97%E7%89%B9%E5%BE%81/">随机变量的数字特征</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%A6%82%E7%8E%87%E8%AE%BA/5.%20%E5%A4%A7%E6%95%B0%E5%AE%9A%E5%BE%8B%E5%8F%8A%E4%B8%AD%E5%BF%83%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86/">大数定律及中心极限定理</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%A6%82%E7%8E%87%E8%AE%BA/6.%20%E6%A0%B7%E6%9C%AC%E5%8F%8A%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/">样本及抽样分布</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%A6%82%E7%8E%87%E8%AE%BA/7.%20%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/">参数估计</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../../%E6%A6%82%E7%8E%87%E8%AE%BA/8.%20%E5%81%87%E8%AE%BE%E9%AA%8C%E8%AF%81/">假设验证</a>
                </li>
    </ul>
                  </li>
                  <li class="toctree-l1 current"><a class="reference internal current" href="#">信号与系统</a>
    <ul class="current">
                <li class="toctree-l2"><a class="reference internal" href="../1.%20%E4%BF%A1%E5%8F%B7%E4%B8%8E%E7%B3%BB%E7%BB%9F/">信号与系统</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../2.%20%E7%BA%BF%E6%80%A7%E6%97%B6%E4%B8%8D%E5%8F%98%E7%B3%BB%E7%BB%9F/">线性时不变系统</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../3.%20%E5%91%A8%E6%9C%9F%E4%BF%A1%E5%8F%B7%E7%9A%84%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0%E8%A1%A8%E7%A4%BA/">周期信号的傅里叶级数表示</a>
                </li>
                <li class="toctree-l2 current"><a class="reference internal current" href="./">连续时间傅里叶变换</a>
    <ul class="current">
    <li class="toctree-l3"><a class="reference internal" href="#_2">非周期信号的表示：连续时间傅里叶变换</a>
        <ul>
    <li class="toctree-l4"><a class="reference internal" href="#_3">非周期信号傅里叶变换表示的导出</a>
    </li>
    <li class="toctree-l4"><a class="reference internal" href="#_4">傅里叶变换的收敛</a>
    </li>
        </ul>
    </li>
    <li class="toctree-l3"><a class="reference internal" href="#_5">周期信号的傅里叶变换</a>
    </li>
    <li class="toctree-l3"><a class="reference internal" href="#_6">连续时间傅里叶变换性质</a>
    </li>
    <li class="toctree-l3"><a class="reference internal" href="#_7">基本傅里叶变换对</a>
    </li>
    <li class="toctree-l3"><a class="reference internal" href="#_8">由线性常系数微分方程表征的系统</a>
    </li>
    </ul>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../5.%20%E7%A6%BB%E6%95%A3%E6%97%B6%E9%97%B4%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2/">离散时间傅里叶变换</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../6.%20%E4%BF%A1%E5%8F%B7%E4%B8%8E%E7%B3%BB%E7%BB%9F%E7%9A%84%E6%97%B6%E5%9F%9F%E5%92%8C%E9%A2%91%E5%9F%9F%E7%89%B9%E6%80%A7/">信号与系统的时域和频域特性</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../7.%20%E9%87%87%E6%A0%B7/">采样</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../9.%20%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%8F%98%E6%8D%A2/">拉普拉斯变换</a>
                </li>
                <li class="toctree-l2"><a class="reference internal" href="../10.%20z%E5%8F%98%E6%8D%A2/">z变换</a>
                </li>
    </ul>
                  </li>
              </ul>
      </div>
    </div>
    </nav>

    <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap">
      <nav class="wy-nav-top" role="navigation" aria-label="Mobile navigation menu">
          <i data-toggle="wy-nav-top" class="fa fa-bars"></i>
          <a href="../..">咩咩的笔记</a>
        
      </nav>
      <div class="wy-nav-content">
        <div class="rst-content"><div role="navigation" aria-label="breadcrumbs navigation">
  <ul class="wy-breadcrumbs">
    <li><a href="../.." class="icon icon-home" aria-label="Docs"></a> &raquo;</li>
          <li>笔记 &raquo;</li>
          <li>信号与系统 &raquo;</li>
      <li>连续时间傅里叶变换</li>
    <li class="wy-breadcrumbs-aside">
    </li>
  </ul>
  <hr/>
</div>
          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
            <div class="section" itemprop="articleBody">
              
                <h1 id="_1">连续时间傅里叶变换</h1>
<h2 id="_2">非周期信号的表示：连续时间傅里叶变换</h2>
<h3 id="_3">非周期信号傅里叶变换表示的导出</h3>
<p>非周期信号可以被看做周期趋于无限的周期信号，于是可以从傅里叶级数推导出<strong>傅里叶变换对</strong>：</p>
<div class="arithmatex">\[
x(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}X(j\omega)e^{j\omega t}d\omega
\]</div>
<div class="arithmatex">\[
X(j\omega)=\int_{-\infty}^{+\infty}x(t)e^{-j\omega t}dt
\]</div>
<p>其中<span class="arithmatex">\(X(j\omega)\)</span>称为<span class="arithmatex">\(x(t)\)</span>的<strong>傅里叶变换</strong>或<strong>傅里叶积分</strong>，而第一条公式称为<strong>傅里叶逆变换</strong>。一个非周期信号<span class="arithmatex">\(x(t)\)</span>的变换<span class="arithmatex">\(X(j\omega)\)</span>通常称为<span class="arithmatex">\(x(t)\)</span>的<strong>频谱</strong></p>
<p>周期信号<span class="arithmatex">\(\tilde{x}(t)\)</span>的傅里叶级数可以利用<span class="arithmatex">\(\tilde{x}(t)\)</span>的一个周期内信号的傅里叶变换的等间隔样本来表示：</p>
<div class="arithmatex">\[
a_k = \frac{1}{T} X(j\omega)\bigg |_{\omega=k\omega_0}
\]</div>
<h3 id="_4">傅里叶变换的收敛</h3>
<p>与周期信号的傅里叶级数类似，有两组充分条件：</p>
<p>第一组条件是能量有限，也就是平方可积：</p>
<div class="arithmatex">\[
\int_{-\infty}^{\infty}|x(t)|^2dt&lt;\infty
\]</div>
<p>同样是可以确保能量上没有任何差别</p>
<p>第二组条件是狄里赫利条件，可以保证除不连续点外任何时刻函数值相等：<br />
1. x(t)绝对可积：<span class="arithmatex">\(\int_{-\infty}^{\infty}|x(t)|dt&lt;\infty\)</span><br />
2. 在任何有限区间内，x(t)只有有限个最大值和最小值<br />
3. 在任何有限区间内，x(t)有有限个不连续点，并且在每个不连续点都必须是有限值  </p>
<p>不过需要补充的是：倘若在变换过程中可以使用冲激函数，那么，在一个无限区间内，既不绝对可积，又不具备平方可积的周期信号也可以认为具有傅里叶变换。</p>
<h2 id="_5">周期信号的傅里叶变换</h2>
<p>可以推导出，周期信号的傅里叶变换可以直接由其傅里叶级数构造，得到的变换在频域由一串冲激组成，各冲激的面积正比于傅里叶级数系数：</p>
<div class="arithmatex">\[
X(j\omega)=\sum_{k=-\infty}^{+\infty}2\pi a_k\delta(\omega-k\omega_0)
\]</div>
<h2 id="_6">连续时间傅里叶变换性质</h2>
<p>以下用<span class="arithmatex">\(x(t)\overset{F}{\leftrightarrow}X(j\omega)\)</span>表示傅里叶变换对</p>
<ul>
<li>线性：<span class="arithmatex">\(ax(t)+by(t)\overset{F}{\leftrightarrow}aX(j\omega)+bY(j\omega)\)</span></li>
<li>时移：<span class="arithmatex">\(x(t-t_0)\overset{F}{\leftrightarrow}e^{-j\omega t_0}X(j\omega)\)</span></li>
<li>频移：<span class="arithmatex">\(e^{j\omega_0 t}x(t)\overset{F}{\leftrightarrow}X(j(\omega-\omega_0))\)</span></li>
<li>共轭：<span class="arithmatex">\(x^*(t)\overset{F}{\leftrightarrow}X^*(-j\omega)\)</span></li>
<li>时间反转：<span class="arithmatex">\(x(-t)\overset{F}{\leftrightarrow}X(-j\omega)\)</span></li>
<li>时域尺度变换：<span class="arithmatex">\(x(at)\overset{F}{\leftrightarrow} \frac{1}{|a|}X(\frac{j\omega}{a})\)</span></li>
<li>卷积：<span class="arithmatex">\(x(t)*y(t)\overset{F}{\leftrightarrow}X(j\omega)Y(j\omega)\)</span></li>
<li>相乘：<span class="arithmatex">\(x(t)y(t)\overset{F}{\leftrightarrow}\frac{1}{2\pi}\int_{-\infty}^{+\infty}X(j\theta)Y(j(\omega-\theta))d\theta\)</span>，也就是“时域相乘，频域卷积”。一个信号被另一个信号相乘，可以被理解为用一个信号去<strong>调制</strong>另一个信号的振幅，因此两个信号相乘往往也称为<strong>幅度调制</strong>，相乘性质被称为<strong>调制性质</strong></li>
<li>微分：<span class="arithmatex">\(\frac{dx(t)}{dt}\overset{F}{\leftrightarrow}j\omega X(j\omega)\)</span></li>
<li>积分：<span class="arithmatex">\(\int_{-\infty}^t x(t)dt\overset{F}{\leftrightarrow}\frac{1}{j\omega}X(j\omega)+\pi X(0)\delta(\omega)\)</span>；频域的冲激项反映了由积分产生的直流或平均值</li>
<li>频域微分：<span class="arithmatex">\(tx(t)\overset{F}{\leftrightarrow}j\frac{d}{d\omega}X(j\omega)\)</span></li>
<li>实信号的共轭对称：当x(t)为实信号时，<span class="arithmatex">\(X(j\omega)=X^*(-j\omega)\)</span></li>
<li>实偶信号：当x(t)是实偶信号时，<span class="arithmatex">\(X(j\omega)\)</span>为实偶函数</li>
<li>实奇信号：当x(t)是实奇信号时，<span class="arithmatex">\(X(j\omega)\)</span>为纯虚奇函数</li>
<li>实信号的奇偶分解：<span class="arithmatex">\(Ev\{x(t)\}\overset{F}{\leftrightarrow}Re\{X(j\omega)\},Od\{x(t)\}\overset{F}{\leftrightarrow}jIm\{X(j\omega)\}\)</span></li>
<li>非周期信号的帕塞瓦尔定理：<span class="arithmatex">\(\int_{-\infty}^{+\infty} |x(t)|^2dt\overset{F}{\leftrightarrow}\frac{1}{2\pi}\int_{-\infty}^{+\infty} |X(j\omega)|^2d\omega\)</span></li>
<li><strong>对偶性质</strong>：由于傅里叶变换和逆变换公式的相似性，已知时域信号到频域函数的变换，就可以简单地通过换元和增加常量的方式，得到与该时域信号形式相同的频域函数对应的时域信号表达式，反之亦然。比如周期信号的频域是冲激串，那么傅里叶变换是周期函数的信号就可以通过对偶性得到其时域是冲激串</li>
</ul>
<h2 id="_7">基本傅里叶变换对</h2>
<table>
<thead>
<tr>
<th>信号</th>
<th>傅里叶变换</th>
<th>傅里叶系数（若为周期的）</th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="arithmatex">\(\sum_{k=-\infty}^{+\infty}a_k e^{jk\omega_0t}\)</span></td>
<td><span class="arithmatex">\(2\pi\sum_{k=-\infty}^{+\infty}a_k\delta(\omega-k\omega_0)\)</span></td>
<td><span class="arithmatex">\(a_k\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(e^{jk\omega_0t}\)</span></td>
<td><span class="arithmatex">\(2\pi\delta(\omega-k\omega_0)\)</span></td>
<td><span class="arithmatex">\(a_1=1,a_k=0,k\neq 1\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(\cos(\omega_0t)\)</span></td>
<td><span class="arithmatex">\(\pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]\)</span></td>
<td><span class="arithmatex">\(a_1=a_{-1}=\frac{1}{2},a_k=0,k\neq\pm1\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(\sin(\omega_0t)\)</span></td>
<td><span class="arithmatex">\(\frac{\pi}{j}[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)]\)</span></td>
<td><span class="arithmatex">\(a_1=-a_{-1}=\frac{1}{2j},a_k=0,k\neq\pm1\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(x(t)=1\)</span></td>
<td><span class="arithmatex">\(2\pi\delta(\omega)\)</span></td>
<td><span class="arithmatex">\(a_0=1,a_k=0,k\neq 0(\forall T)\)</span></td>
</tr>
<tr>
<td>工作周期为<span class="arithmatex">\(2T_1\)</span>周期为<span class="arithmatex">\(T\)</span>的中心对称方波</td>
<td><span class="arithmatex">\(\sum_{k=-\infty}^{+\infty}\frac{2\sin(k\omega_0T_1)}{k}\delta(\omega-k\omega_0)\)</span></td>
<td><span class="arithmatex">\(\frac{\omega_0T_1}{\pi}sinc(\frac{k\omega_0T_1}{\pi})=\frac{\sin(k\omega_0T_1)}{k\pi}\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(\sum_{k=-\infty}^{+\infty}\delta(t-nT)\)</span></td>
<td><span class="arithmatex">\(\frac{2\pi}{T}\sum_{k=-\infty}^{+\infty}\delta(\omega-\frac{2\pi k}{T})\)</span></td>
<td><span class="arithmatex">\(a_k=\frac 1 T\)</span></td>
</tr>
<tr>
<td>长度为<span class="arithmatex">\(2T_1\)</span>的中心对称单周期方波</td>
<td><span class="arithmatex">\(\frac{2\sin(\omega T_1)}{\omega}\)</span></td>
<td>-</td>
</tr>
<tr>
<td><span class="arithmatex">\(\frac{\sin(Wt)}{\pi t}\)</span></td>
<td>长度为2W的中心对称单周期方波</td>
<td>-</td>
</tr>
<tr>
<td><span class="arithmatex">\(\delta(t)\)</span></td>
<td><span class="arithmatex">\(1\)</span></td>
<td>-</td>
</tr>
<tr>
<td><span class="arithmatex">\(u(t)\)</span></td>
<td><span class="arithmatex">\(\frac{1}{j\omega}+\pi \delta(\omega)\)</span></td>
<td>-</td>
</tr>
<tr>
<td><span class="arithmatex">\(\delta(t-t_0)\)</span></td>
<td><span class="arithmatex">\(e^{-j\omega t_0}\)</span></td>
<td>-</td>
</tr>
<tr>
<td><span class="arithmatex">\(e^{-\alpha t}u(t),Re\{\alpha\}&gt;0\)</span></td>
<td><span class="arithmatex">\(\frac{1}{\alpha+j\omega}\)</span></td>
<td>-</td>
</tr>
<tr>
<td><span class="arithmatex">\(te^{-\alpha t}u(t),Re\{\alpha\}&gt;0\)</span></td>
<td><span class="arithmatex">\(\frac{1}{(\alpha+j\omega)^2}\)</span></td>
<td>-</td>
</tr>
<tr>
<td><span class="arithmatex">\(\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t),Re\{\alpha\}&gt;0\)</span></td>
<td><span class="arithmatex">\(\frac{1}{(\alpha+j\omega)^n}\)</span></td>
<td>-</td>
</tr>
</tbody>
</table>
<h2 id="_8">由线性常系数微分方程表征的系统</h2>
<p>对于满足如下形式的线性常系数微分方程的系统：</p>
<div class="arithmatex">\[
\sum_{k=0}^{N}a_k\frac{d^ky(t)}{dt^k}=\sum_{k=0}^{M}b_k\frac{d^kx(t)}{dt^k}
\]</div>
<p>可以通过如下的两个多项式之比求出其频率响应：</p>
<div class="arithmatex">\[
H(j\omega)=\frac{\sum_{k=0}^{M}b_k(j\omega)^k}{\sum_{k=0}^{N}a_k(j\omega)^k}
\]</div>
<p>通过待定系数将频率响应拆成<span class="arithmatex">\(\frac{C}{(\alpha+j\omega)^n}\)</span>的和，即可得到若干个<span class="arithmatex">\(C\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)\)</span>的和的时域信号</p>
              
            </div>
          </div><footer>
    <div class="rst-footer-buttons" role="navigation" aria-label="Footer Navigation">
        <a href="../3.%20%E5%91%A8%E6%9C%9F%E4%BF%A1%E5%8F%B7%E7%9A%84%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0%E8%A1%A8%E7%A4%BA/" class="btn btn-neutral float-left" title="周期信号的傅里叶级数表示"><span class="icon icon-circle-arrow-left"></span> Previous</a>
        <a href="../5.%20%E7%A6%BB%E6%95%A3%E6%97%B6%E9%97%B4%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2/" class="btn btn-neutral float-right" title="离散时间傅里叶变换">Next <span class="icon icon-circle-arrow-right"></span></a>
    </div>

  <hr/>

  <div role="contentinfo">
    <!-- Copyright etc -->
  </div>

  Built with <a href="https://www.mkdocs.org/">MkDocs</a> using a <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>.
</footer>
          
        </div>
      </div>

    </section>

  </div>

  <div class="rst-versions" role="note" aria-label="Versions">
  <span class="rst-current-version" data-toggle="rst-current-version">
    
    
      <span><a href="../3.%20%E5%91%A8%E6%9C%9F%E4%BF%A1%E5%8F%B7%E7%9A%84%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0%E8%A1%A8%E7%A4%BA/" style="color: #fcfcfc">&laquo; Previous</a></span>
    
    
      <span><a href="../5.%20%E7%A6%BB%E6%95%A3%E6%97%B6%E9%97%B4%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2/" style="color: #fcfcfc">Next &raquo;</a></span>
    
  </span>
</div>
    <script>var base_url = '../..';</script>
    <script src="../../js/theme_extra.js" defer></script>
    <script src="../../js/theme.js" defer></script>
      <script src="../../javascripts/mathjax.js" defer></script>
      <script src="https://fastly.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" defer></script>
      <script src="../../search/main.js" defer></script>
    <script defer>
        window.onload = function () {
            SphinxRtdTheme.Navigation.enable(true);
        };
    </script>

</body>
</html>
